How do you use the half-angle identity to find the exact value of csc 112.5?

1 Answer
Nghi N.
Aug 21, 2015

Find #csc (112.5)#

Ans: #2/(sqrt(2 + sqrt2)#

Explanation:

#csc (112.5) = 1/sin (112.5)#. Call sin 112.5 = sin t
#cos 2t = cos 225 = cos (45 + 180) = -cos 45 = -sqrt2/2#
Apply the trig identity: #cos 2t = 1 - 2sin^2 t#
#-sqrt2/2 = 1 - 2sin^2 t#
#sin^2 t = (2 + sqrt2)/4#

#sin (112.5) = sin t = sqrt(2 + sqrt2)/2.#
Only the positive number is accepted because the arc 112.5 is located in Quadrant II.

Finally, #csc 112.5 = 1/sin 112.5 = 2/sqrt(2 + sqrt2)#
Check by calculator, sin 112.5 = 0.923--> csc 112.5 = 1.08
#2/(sqrt(2 + sqrt2)# = #2/sqrt(3.414) = 2/1.85 = 1.08# OK

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